In abstract algebra, a finite field or Galois field is a field that contains a finite number of elements. Arithmetic in a finite field is different from standard integer arithmetic. There are a limited number of elements in the finite field and all operations performed in the finite field result in an element within that field.
While each finite field is itself not infinite, there are infinitely many different finite fields. The number of elements also called cardinality in a finite field is of the form p where p is a prime number and n is a positive integer. Furthermore, two fields of the same size are isomorphic. The prime p is called the characteristic of the field, and the positive integer n is called the dimension or size of the field.
The Galois Field having characteristic 2 and size 1 is commonly denoted as GF(2). Polynomials over the GF(2) field have coefficients which are elements of the GF(2) field. Moreover, arithmetic over the GF(2) field and polynomial arithmetic over the GF(2) field are the basis for many computer applications. For example, polynomial arithmetic over the GF(2) field is used to produce error correcting codes such as, for example, BCH error-correcting codes and Reed Solomon error correcting codes. Polynomial arithmetic over the GF(2) field is also used in digital signal processing applications such as in determining an Infinite Impulse Response (IIR). Furthermore, polynomial arithmetic over the GF(2) field is used in cryptography algorithms such as the Advanced Encryption Standard (AES) block cipher. Thus, efficient polynomial arithmetic over the GF(2) field may have a positive effect on performance of a computing device that utilizes such error correcting codes and/or cryptographic algorithms.